WASHINGTON—The Department of Veterans Affairs will replace the acting head of its health care system with a temporary appointee on Wednesday, the VA’s acting secretary said Tuesday, though the department still hasn’t settled on a permanent replacement to lead the sprawling division.

Dr. Richard Stone will take over the Veterans Health Administration, the VA’s acting secretary, Peter O’Rourke, told reporters. The health administration is one of the three main branches of the VA and comprises more than 1,500 health-care facilities,...